A weak second term identity of the regularized Siegel-Weil formula for unitary groups

TL;DR

This paper establishes a weak second term identity for the regularized Siegel-Weil formula in the context of unitary groups, extending previous work and deriving a Rallis inner product formula for theta lifts.

Contribution

It introduces a new weak second term identity for the regularized Siegel-Weil formula for certain unitary dual pairs, generalizing prior results by Gan and Takeda.

Findings

Derived a weak second term identity for the regularized Siegel-Weil formula.

Established a Rallis inner product formula for theta lifts from U(W) to U(V).

Extended the understanding of the Siegel-Weil formula in the unitary group setting.

Abstract

Following W.T.Gan and S.Takeda, we obtain a weak second term identity of the regularized Siegel-Weil formula for the unitary dual pair $(U(n,n),U(V))$, where $V$ is a split hermitian space of dimension $2r$ with $r+1 \leq n \leq 2r-1$. As an application, we obtain a Rallis inner product formula for theta lifts from $U(W)$ to $U(V)$ for a skew-hermitian space $W$ of dimension $n$.